L3
. |
-
[
§

RIS e T RS S AR

B At B BT R AT L T
" ahall 4 TRTRTRTRIE 1o ok R BOOE Y PR AT e

36 THE FASCINATION OF NUMBERS

clue to this assumption was given by the fact that the last two
digits of 93 (=729) are 29 and this equals (3 xg) +2 whilst
the last three digits of 992 (=970299) are 299 and this equals
(3 X99) +2. It is then necessary to show how the remaining
portion of each cube number is built up (that is, the num-
bers in the series o, 16, 50, 108, etc., shown above).

For the cube of 4, this number is 50 and this equals 2(5)2.
For the cube of 5, the relevant number is 108 and this equals
3(6)2 These two examples at once reveal their pattern, and
we now have

4*=2(5)"+(3)(4) +2
giving the general equation:
¥3=(x—2)(x4+1)2+43x+2
which may be proved algebraically since (x—2)(x+1)?2
equals ¥ —gx—2.

This is not a useful equation for evaluating cube numbers
in general, but it renders the particular cases of ‘all-g’ cubes
very simple indeed.

999°=(997) (1000)*+3(999) +2
=097,000,000 +2997 -2
=997,002,999.
It can also be used for certain other selected numbers.
49°=(47)(50)*+3(49) +2
=I117,500 —147 -+2
=117,649.

Here it may be mentioned that although the powers of
numbers consisting entirely of nines are very much of a pat-
tern, so that one answer can be derived from another merely
by the insertion of other digits, this does not result from any
magic power possessed by the digit g. It results instead from
the fact that g is one less than 10; g9 is one less than 100, and
SO on.

The difference between any two consecutive cube num-
bers may be found by a simplified process. The difference
between 102 and 9? is 271 (that is, 1000 —7%29) and this dif-